Problem: Find the number of real solutions of the equation
\[\frac{x}{100} = \sin x.\]
Explanation: Since $-1 \le \sin x \le 1,$ all solutions must lie in the interval $[-100,100].$

[asy]
unitsize (1 cm);

real func (real x) {
  return (2*sin(pi*x));
}

draw(graph(func,0,4.2),red);
draw(graph(func,8.8,12),red);
draw((0,0)--(4.5,2/11.8*4.5),blue);
draw((8.8,2/11.8*8.8)--(11.8,2),blue);
draw((0,-2)--(0,2));
draw((0,0)--(12,0));

draw((1,-0.1)--(1,0.1));
draw((2,-0.1)--(2,0.1));
draw((3,-0.1)--(3,0.1));
draw((4,-0.1)--(4,0.1));
draw((9,-0.1)--(9,0.1));
draw((10,-0.1)--(10,0.1));
draw((11,-0.1)--(11,0.1));
draw((12,-0.1)--(12,0.1));

label("$\pi$", (1,-0.1), S, UnFill);
label("$2 \pi$", (2,-0.1), S, UnFill);
label("$3 \pi$", (3,-0.1), S, UnFill);
label("$4 \pi$", (4,-0.1), S, UnFill);
label("$29 \pi$", (9,-0.1), S, UnFill);
label("$30 \pi$", (10,-0.1), S, UnFill);
label("$31 \pi$", (11,-0.1), S, UnFill);
label("$32 \pi$", (12,-0.1), S, UnFill);
label("$\dots$", (13/2, 1));

label("$y = f(x)$", (13,-1), red);
label("$y = \frac{x}{100}$", (11.8,2), E, blue);
[/asy]

Note that $\frac{100}{\pi} \approx 31.83.$  This means that when the graph of $y = \sin x$ reaches 1 at $x = \left( 30 + \frac{1}{2} \right) \pi,$ this point lies above the line $y = \frac{x}{100},$ and that this is the last crest of the sine function that intersects the line $y = \frac{x}{100}.$

We see that on the interval $[2 \pi k, 2 \pi (k + 1)],$ where $0 \le k \le 15,$ the graphs of $y = \frac{x}{100}$ and $y = \sin x$ intersect twice.  Thus, there are $2 \cdot 16 = 32$ solutions for $0 \le x \le 100.$  By symmetry, there are also 32 solutions for $-100 \le x \le 0,$ but this double-counts the solution $x = 0.$  Thus, there are a total of $32 + 32 - 1 = \boxed{63}$ solutions.